The Levitating Superconductor Ring · AI Generator #GEN-0460527F

Physics · electromagnetism

Problem

A thin superconducting ring of radius

R = 5.0\;\text{cm}

and mass

m = 15\;\text{g}

is placed around a vertical solenoid. The solenoid has

N = 1000

turns per meter and carries a current that increases linearly with time according to

I(t) = \alpha t

, where

\alpha = 2.0\;\text{A/s}

. Due to the changing magnetic flux, an induced current flows in the superconducting ring, creating a magnetic force that can levitate the ring against gravity. At what height

h

above the bottom of the solenoid will the ring hover in equilibrium? Assume the magnetic field inside the solenoid is uniform and equal to

B = \mu_0 n I

, where

n

is the turn density. Outside the solenoid, assume the field falls off as

B(z) = B_0 \frac{R_s^2}{(R_s^2 + z^2)^{3/2}}

where

R_s = 2.0\;\text{cm}

is the solenoid radius,

z

is the distance from the solenoid end, and

B_0

is the field at the solenoid end.

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